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 Incoming links: Half-line advection-diffusion global relation

Rapid decay allows two integrations by parts in the half-line Fourier transform. They give

\partial_{t} u + ω u = - u_{x} (0, t) - (ik + β) u (0, t) .

Solving this scalar inhomogeneous equation by an integrating factor yields

u (k, t) = e^{- ω t} u (k, 0) - (ik + β) f_{0} (ω, t) - f_{1} (ω, t) .

The initial transform is holomorphic for

Im k < 0

and continuous on the real axis under the given decay. The time integrals are entire functions of

ω

for fixed

t

and smooth boundary data.